Beta kernel estimators for density functions

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Abstract

Kernel estimators using non-negative kernels are considered to estimate probability density functions with compact supports. The kernels are chosen from a family of beta densities. The beta kernel estimators are free of boundary bias, non-negative and achieve the optimal rate of convergence for the mean integrated squared error. The proposed beta kernel estimators have two features. One is that the different amount of smoothing is allocated by naturally varying kernel shape without explicitly changing the value of the smoothing bandwidth. Another feature is that the support of the beta kernels can match the support of the density function; this leads to larger effective sample sizes used in the density estimation and can produce density estimates that have smaller finite-sample variance than some other estimators.

Introduction

Let X1,…,Xn be a random sample from a distribution with an unknown probability density function f. A standard kernel density estimator for f isf̂(x)=(nh)−1∑K{h−1(x−Xi)},where K and h are the kernel function and the smoothing bandwidth, respectively. Comprehensive reviews of the kernel smoothing method are available in Silverman (1986), Scott (1992) and Wand and Jones (1995). The standard kernel estimator (1.1) was developed primarily for densities with unbounded supports. The kernel function K is usually symmetric and is regarded as less important than the smoothing bandwidth. While using a symmetric kernel is appropriate for fitting densities with unbounded supports, it is not adequate for densities with compact supports as it causes boundary bias.

Removing boundary bias has been an active area of research. Data reflection was proposed by Schuster (1985), but is only effective when the density functions have a zero derivative at boundary points. Boundary kernels were suggested by Müller (1991). Cowling and Hall (1996) proposed generating pseudodata beyond the boundary points by linear interpolation of the order statistics. Marron and Ruppert (1994) considered using empirical transformations. In recent years, the local polynomial regression method of Cleveland (1979) has been applied to density estimation by Lejeune and Sarda (1992) and Jones (1993). Another version of the local linear estimator was considered by Cheng et al. (1996) by data binning and a local polynomial fitting on the bin counts. As the local linear estimators can have negative values (so too the boundary kernel estimators), Jones and Foster (1996) proposed a non-negative estimator by combining a local linear estimator and a re-normalizing kernel estimator.

In the simple spirit of the standard kernel estimator (1.1), this paper considers estimators using a beta family of density functions as kernels to estimate density functions with compact supports. The idea of smoothing using the beta kernels has been discussed in Chen (1999) in the context of non-parametric regression. This paper concentrates on density estimation and finite-sample comparisons with the local linear estimator of Lejeune and Sarda (1992) and Jones (1993) and its non-negative modification proposed by Jones and Foster (1996). The beta kernel method is easy both in concept and in its implementation, and produces estimators which are free of boundary bias, always non-negative and which have mean integrated squared errors of O(n−4/5). There are two special features about the beta kernels. One is that the shape of the beta kernels varies naturally, which leads to the amount of smoothing being changed according to the position where the density estimation is made without explicitly changing the bandwidth; this implies that the beta kernel estimators are adaptive density estimators. The other feature is that the support of the beta kernels matches the support of the density function; this leads to larger effective sample size used in the density estimation and can produce density estimates that have smaller variance.

The paper is structured as follows. In Section 2 beta kernel estimators are introduced. Their bias and variance properties are studied in Section 3. In Section 4, the mean integrated squared errors and the optimal smoothing bandwidths are derived. Section 5 compares the variance of the beta kernel estimators with that of a local linear estimator. The beta kernel estimators are used in Section 6 to analyze a tuna data set as an example. Section 7 presents results from a simulation study.

Section snippets

Beta kernel estimators

The idea of beta kernel smoothing was motivated by the Bernstein theorem in mathematical function analysis, as explained in Brown and Chen (1999) when considering estimation of regression curves with equally spaced fixed design points by combining the beta kernels and Bernstein polynomials. The Bernstein theorem states that for any function which is continuous and has a bounded support its Bernstein polynomials converge to the function uniformly. The uniform convergence means the Bernstein

Bias and variance

Note thatE{f̂1(x)}=01Kx/b+1,(1−x)/b+1(y)f(y)dy=E{f(ξx)},where ξx is a Beta{x/b+1,(1−x)/b+1} random variable. Using the same derivation for the bias of the beta kernel regression estimator given in Chen (1999), it can be shown that the bias of the beta estimator isBias{f̂1(x)}={(1−2x)f′(x)+12x(1−x)f″(x)}b+o(b),where the remainder term is uniformly o(b) for x∈[0,1]. The bias is of O(b) throughout [0,1], indicating that f̂1 is free of boundary bias.

The involvement of f′ in the bias is due to the

Global properties

Let δ=b1−ε where 0<ε<1. From (3.5), for i=1 or 2,01Var{f̂i(x)}dx=0δ+δ1−δ+1−δ1Var{f̂1(x)}dx=δ1−δ12π{x(1−x)}−1/2n−1b−1/2f(x)dx+O(n−1b−ε)=12πn−1b−1/201{x(1−x)}−1/2f(x)dx+o(n−1b−1/2)by choosing ε properly.

Combining , , , the mean integrated squared errors for f̂1 and f̂2 are, respectively,MISE(f̂1)=b201{(1−2x)f′(x)+12x(1−x)f″(x)}2dx+12πn−1b−1/201{x(1−x)}−1/2f(x)dx+o(n−1b−1/2+b2)andMISE(f̂2)=14b201{x(1−x)f″(x)}2dx+12πn−1b−1/201{x(1−x)}−1/2f(x)dx+o(n−1b−1/2+b2).

The optimal bandwidths which

Variance comparison

Even though the variance behaviour of the beta kernel estimators near the boundaries has little effect on the mean integrated squared error, one may still be concerned about it. To allay this concern, in this section we compare the variance of the beta kernel estimators with that of the local linear estimator of Lejeune and Sarda (1992) and Jones (1993) and the non-negative estimator of Jones and Foster (1996).

For non-negative integers s and m and any symmetric kernel K with compact support

An example

We first apply the proposed beta kernel smoothers to a tuna data set, given in Chen (1996), which were collected from an aerial line transect survey to estimate the abundance of Southern Bluefin Tuna over the Great Australian Bight. The data are the perpendicular sighting distances (in miles) of detected tuna schools to the transect lines flown by a light airplane with tuna spotters on board. The tuna abundance estimation relies on the estimation of f, the probability density function of the

Simulation results

In this section we report results of a simulation study designed to investigate performance of the proposed beta kernel estimators f̂1 and f̂2. For comparison purposes, the local linear estimator of Jones (1993) and the non-negative estimator of Jones and Foster (1996) were also evaluated. It should be pointed out that a comparison or competition among various estimators is, at times, hardly fair. For instance, in the current comparison, the non-negative estimator is constructed using two

Acknowledgements

The author thanks Professor B.M. Brown for stimulating discussions, Professor M.C. Jones for constructive comments on an early version of the paper and a referee for bringing up some previous works on beta kernel smoothing into his attention.

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